物理学中的拓扑与几何

1  Introduction

References

2  Topology

2.1  Basic Definitions

2.2  Base of Topology, Metric, Norm

2.3  Derivatives

2.4  Compactness

2.5  Connect

1 Introduction

References

2 Topology

2.1 Basic Definitions

2.2 Base of Topology, Metric, Norm

2.3 Derivatives

2.4 Compactness

2.5 Connectedness, Homotopy

2.6 Topological Charges in Physics

References

3 Manifolds

3.1 Charts and Atlases

3.2 Smooth Manifolds

3.3 Tangent Spaces

3.4 Vector Fields

3.5 Mappings of Manifolds, Submanifolds

3.6 Frobenius' Theorem

3.7 Examples from Physics

3.7.1 Classical Point Mechanics

3.7.2 Classical and Quantum Mechanics

3.7.3 Classical Point Mechanics Under Momentum Constraints

3.7.4 Classical Mechanics Under Velocity Constraints

3.7.5 Thermodynamics

References

4 Tensor Fields

4.1 Tensor Algebras

4.2 Exterior Algebras

4.3 Tensor Fields and Exterior Forms

4.4 Exterior Differential Calculus

References

5 Integration, Homology and Cohomology

5.1 Prelude in Euclidean Space

5.2 Chains of Simplices

5.3 Integration of Differential Forms

5.4 De Rham Cohomology

5.5 Homology and Homotopy

5.6 Homology and Cohomology of Complexes

5.7 Euler's Characteristic

5.8 Critical Points

5.9 Examples from Physics

References

6 Lie Groups

6.1 Lie Groups and Lie Algebras

6.2 Lie Group Homomorphisms and Representations

6.3 Lie Subgroups

6.4 Simply Connected Covering Group

6.5 The Exponential Mapping

6.6 The General Linear Group Gl(n,K)

6.7 Example from Physics: The Lorentz Group

6.8 The Adjoint Representation

References

7 Bundles and Connections

7.1 Principal Fiber Bundles

7.2 Frame Bundles

7.3 Connections on Principle Fiber Bundles

7.4 Parallel Transport and Holonomy

7.5 Exterior Covariant Derivative and Curvature Form

7.6 Fiber Bundles

7.7 Linear and Affine Connections

7.8 Curvature and Torsion Tensors

7.9 Expressions in Local Coordinates on M

References

8 Parallelism, Holonomy, Homotopy and (Co)homology

8.1 The Exact Homotopy Sequence

8.2 Homotopy of Sections

8.3 Gauge Fields and Connections on R4

8.4 Gauge Fields and Connections on Manifolds

8.5 Characteristic Classes

8.6 Geometric Phases in Quantum Physics

8.6.1 Berry-Simon Connection

8.6.2 Degenerate Case

8.6.3 Electrical Polarization

8.6.4 Orbital Magnetism

8.6.5 Topological Insulators

8.7 Gauge Field Theory of Molecular Physics

References

9 Riemannian Geometry

9.1 Riemannian Metric

9.2 Homogeneous Manifolds

9.3 Riemannian Connection

9.4 Geodesic Normal Coordinates

9.5 Sectional Curvature

9.6 Gravitation

9.7 Complex, Hermitian and K~ihlerian Manifolds

References

Compendium

List of Symbols

Index

《物理学中的拓扑与几何(英文影印版)》讲述了在物理学中应用的拓扑和几何知识，包括流形、张量场、流形上的微积分、纤维丛理论等。特别地，本书讲解了这些理论在物理学中的诸多应用。
　　随着理论物理的发展，拓扑与几何这些数学理论在物理中的应用日益广泛。特别地，在理论物理近些年的一些新理论中，拓扑和几何的应用更加重要。本书系统而深入，其引进能够给理论物理工作者以很大帮助。

数学与物理学的结合，从来就非常紧密。没有微积分的发明，不会有经典物理学的蓬勃发展。现代物理学依然与数学存在着极为重要的联系，并且在发展中互相促进。现在，无论是引力、场论、粒子物理，或是凝聚态物理理论的研究者，一定的拓扑与微分几何知识都是必须具备的，更不用说像超弦理论之类的新物理了。本书全面而深入浅出地讲述了这些物理学中所用到的拓扑与几何知识，是理论物理研究者非常难得的参考书，有志于理论物理研究的读者千万不能错过。